arXiv:astro-ph/0010026v1 2 Oct 2000 rpittpstuigL using typeset Preprint vlto.Tetmeaueo h eta tro h ei h post-AGB Helix its the H of of the track by central cooling measured the been the of down temperature The well is evolution. it that cating re- ionized the that disk. suggested a has is 1982) (1998) gion White O’Dell & (Meaburn made. elli ones being both i bipolar of and these suggestions with shapes of difficult, cal/toroidal deprojection be The to proven III]. has [O ages in shape elliptical more 2 o40p.Otcliae hwacmlctdmorphol- H complicated in a structure show helical images a Optical fro by ranging characterized pc. values ogy giving 400 PN, to t this applied 120 to been have distance methods Other the 1997). determine al. et (Harris pc 1997). 213 al. et (Young I ou least C at in and, nebula, 1992) main al. the et side (Huggins CO hav in knots detected The been kno also appearance. resolved symmetric spatially highly & with of Burkert 3500, multitude about 1997; a Burkert revealed & 1998) 1996, O’Dell O’Dell 1996; 1992, observations Handron HST Camera & al. Planetary cen- (O’Dell et and the Field (Meaburn Wide from work and away 1998) Groundbased a pointing tails have source. their tral 1968) Vorontsov–Velyaminov with shape, by (which knots cometary reported These knots. first small were of thousands many by terized nw steHlxNbl NC79,P 36-57 PK 7293, (NGC Nebula Helix the as known hrce ODl 98 er,Kitr&Dfu 99.The 1999). ionizati Dufour low & ionizatio Kwitter and complicated Henry, high a 1998; to of (O’Dell diam- leads regions character in pc, pc co-existing 0.5 213 with about of structure , distance PN a the at of eter size large the and luminosity Gry Stasi (Górny, D ATVERSION RAFT h eta tri eyhtadhsalwlmnst,indi- luminosity, low a has and hot very is star central The hsP soeo h lss ou ihaprla itneof distance parallax a with us to closest the of one is PN This h ml cl tutr ftePaeayNbl (PN) Nebula Planetary the of structure scale small The h ihefcietmeauecmie ihtefil low fairly the with combined temperature effective high The 3 2 1 nttt eAtooí,UA,J .Tbaa10,ClnaL Colonia 1006, Tablada J. J. UNAM, Astronomía, de Instituto Sweden Saltsjöbaden, 36 S-133 Observatory, Stockholm nttt eAtooí,UA,Ad.Psa 024 41 M 04510 70-264, Postal Apdo. UNAM, Astronomía, de Instituto H AAACIEA H PC EECP CEC INSTITUTE. SCIENCE TELESCOPE SPACE THE AT ARCHIVE DATA THE aito ftecnrlsa.W opr hs oeswt t with models these compare We ph star. a central as globules the the of of head radiation the of emission the describes that pc Telescope Space fteaayia oe eoti aeo oiigpoosf photons ionizing of rate a obtain we model analytical the of ossetwt siae ae ntettlH total the on based estimates with consistent fti oe ihtevle eemndfo h observatio the from determined headings: values Subject stell the the through with direct model data to this observational diffuse of the of from a ratio obtained obtain the to those with us compare produ allows We images region HST flux. shadow the ionizing neutral with a comparison A from nebula. wind photoevaporated a as HTEAOAIGFOSFO H OEAYKOSI H HELIX THE IN KNOTS COMETARY THE FROM PHOTOEVAPORATING FLOWS eepanteH the explain We a O ATAL AE NOSRAIN AEWT H AAEAHUBB NASA/ESA THE WITH MADE OBSERVATIONS ON BASED PARTIALLY sa&Tlna1997). Tylenda & nska ´ CTOBER A T E tl mltajv 04/03/99 v. emulateapj style X 9 2018 29, 1. β INTRODUCTION rhvliae fteHlxNbl.Fo oprsno th of comparison a From Nebula. Helix the of images archival ) asr ehdt be to method Zanstra S:srcue—paeaynble niiul(G 23 — 7293) (NGC individual nebulae: planetary — structure ISM: .L L. α msino h oeaykosi h ei eua(G 7293) (NGC Nebula Helix the in knots cometary the of emission ÓPEZ -M ARTÍN α 1 n NI]ada and II] [N and .C R C. A. , ≃ ◦ )i charac- is 1) 1 rf eso coe 9 2018 29, October version Draft . 2 msd at aí,500Mrla ihaá,México Michoacán, Morelia, 58090 María, Santa de omas × β xc,D . México F., D. éxico, AGA 10 u ftenbl.W lomdltetiso h oeayknot cometary the of tails the model also We nebula. the of flux ABSTRACT pti- 5 m- 1 on ts, as m .M G. , K t- o n e 1 ELLEMA u ftepoovprtdwn nasmlrwyt Mellema to way similar a par in the wind and some photoevaporated flux derive the stellar of we ionizing flux First, the (PN). relate Nebula that expressions Planetary a in embedded eta lmslctdi h ue at fapaeaynebu planetary a of nebula). den parts Helix of outer the evolution the (like the in study located (1998) external clumps dyna al. neutral an and et thermal Mellema by including effects. gas cal radiation of ultraviolet modelled clumps of (1998) dense source Bally of & Hollenbach the con Johnstone, cloud cometary equilibrium o figuration. an radiation finding star, ionizing formed the newly to o evolution exposed the cloud of photoevaporating theory (1990 analytic McKee & approximate Bertoldi an field. developed radiation ionizing an in clumps xaso eoiiso h nt n h Orn ob 9and 19 be to ring CO the s km and deproject 29 knots the the the found of (1999) follow velocities al. expansion knots et typ Young lower the a emission. velocity. at that although CO expansion CO, shown the scale as have the large distribution velocity (1998) in the same neutral found al. these in also of et and is form Meaburn knots PN the the the in of of substanti is shape distribution a PN aspherical that the The result of curious globules. mass the the to th of leads fraction for This derived 1996). densities dron the with globules, contrast neutral/molecular in 1998), (O’Dell tl aeadnmclefc nteP structure. PN the on effect dynamical a have still aigi ital neetbe oee,wt t expect its with ( However, velocity win density high undetectable. low virtually very a it implies also making star the of luminosity low teaoae o rdcdb h nietionizing incident the by produced flow otoevaporated nltclti n edwn oe.Teagreement The model. wind head and tail analytical 2. so h nt sexcellent. is knots the of ns rflxa ucino aisisd nHIregion HII an inside radius of function a as flux ar o h eta tro bu 5 about of star central the rom e sdsusasml,aayia oe o oeayknots cometary for model analytical simple, a discuss us Let oeohratoshv rae h hteaoainof photoevaporation the treated have authors other Some h esrddniisi h euaaelw bu 0cm 60 about low, are nebula the in densities measured The eH he NLTCLMDLFRTEPOOVPRTDFU OF PHOTOEVAPORATED FLUX THE FOR MODEL ANALYTICAL e ytedfueinzn htnfil fthe of field photon ionizing diffuse the by ced 2 ietdtriaino h au ftediffuse the of value the of determination direct .J H J. W. , − α 1 respectively. , msinotie rmteHT( HST the from obtained emission ESAETLSOE BANDFROM OBTAINED TELESCOPE, SPACE LE v ENNEY ∞ ≃ H e 00k s km 6000 H OEAYKNOTS COMETARY THE 3 α .C J. , tr:ABadpost-AGB and AGB : msinwt h predictions the with emission ANTÓ n EUA(G 7293) (NGC NEBULA iha nltclmodel analytical an with knot − 1 × ∼ 1 frdaindie)i could it driven) radiation if 10 10 45 6 cm s − 1 − hc is which , 3 Hubble ODl Han- & (O’Dell a s ticle ical mi- a f a f ed ed se − d, al la e 3 - ) 2 PhotoevaporatingflowsfromthecometaryknotsintheHelix nebula (NGC 7293)

θ

Rh

F * n h F h h

FIG. 1.— Schematic diagram of the cometary head. Fh is the incident flux at the surface of the neutral knot, and Rh and nh are the radius and the density of the neutral knot (respectively). et al. (1998) to find the Hα emission for the cometary heads. + 2 Cantó et al. (1998) proposed the idea of the tails as neutral F⋆ cos(θ) ≃ Fh(θ) hnh(θ) αB , (3) shadow regions behind the clumps, and studied the complex time-evolution of the resulting flow. In this paper, we model the where αB is the case B recombination coefficient of hydrogen, tails behind the Helix clumps as a cylinder of neutral material which is assumed to be constant with position. The flow thick- being photoionized by the diffuse ionizing flux of the nebula. ness h is defined by : h ≡ R (4) 2.1. A model for the cometary head ω h , Let us consider the problem of a hemispherical, neutral knot where the parameter ω is defined through the relation : of radius Rh which is being photoionized by the radiative field ∞ 2 2 emitted by the central star of the ionized nebula. A schematic ω nh Rh ≡ n (r) dr . (5) diagram of this configuration is shown in Figure 1. ZRh A fraction of the stellar flux arrives at the knot surface, pho- This parameter ω depends on the density profile of the photo- toionizing the neutral knot material and feeding a photoevapo- evaporated wind. For a D-critical ionization front (v(Rh)= ci, rated flow. The remaining ionizing photons are absorbed in this where ci is the isothermal sound speed of the ionized gas), photoevaporated flow. The neutral knot is being photoevapo- ω ≃ 0.1 (Henney & Arthur 1998). From equations (1-5) we rated, but we consider a quasi-steady state in which the radius then obtain a relation between the ionizing flux from the cen- of the knot Rh is almost constant with time. The incident flux at tral star F⋆, and the incident flux at the knot surface Fh(θ) : the knot surface is : R ≃ + 2 ω h αB F⋆ cos(θ) Fh(θ) Fh (θ) 2 , (6) Fh(θ)= nh(θ) v(Rh), (1) ci where Fh is the flux of ionizing photons incident normal to the The term on the left is the ionizingstellar flux as a functionof knot surface (per unit area and time), nh and v(Rh) are the den- the angle of incidence, the first term on the right is the incident sity and the velocity of the photoevaporated wind at the knot ionizing flux on the knot surface that produces the photoion- surface . We note that the rate of photoionizations depends izations of the neutral material and the second term is the flux on the angle of incidence θ (see Figure 1), having a maximum absorbed in the photoevaporated wind. If we define : value in the front of the cometary head. 2 In order to obtain the density profile, we consider the conser- ci ξh ≡ , (7) vation of particles in the photoevaporated wind : Rh ω αB equation (6) can be written as : 2 2 R n(R,θ)v(R)= Rhnh(θ)v(Rh), (2) 2 Fh (θ) where R is the radius directed outwards from the knot. F⋆ cos(θ) ≃ Fh(θ) + , (8) We assume that the ionization front is thin compared with the ξh effective thickness h of the ionized flow and that h in turn is We can see from equation (8) that it is possible to find the small compared with Rh. In this case, if all diffuse photons are incident flux Fh(θ) as a function of the ionizing flux F⋆ solving absorbed “on-the-spot”, the photoionization balance is given by the quadratic equation to obtain : López-Martín et al. 3

1/2 ξ 4 F cos(θ) Ft = nt ci , (15) F (θ) ≃ h ⋆ + 1 − 1 . (9) h 2 ξ " h  # Defining the parameter :

We find two different limiting situations : 2 ′′ ci If ξh/cos(θ) << F⋆ we have the Recombination Dominated ξt ≡ , (16) Regime ′′ in which the fluxes are related by the expression : ω αB Rt and subsituting equations (15-16) in (14) one obtains : Fh(θ) ≈ ξh F⋆ cos(θ), (10) 2 In this case, an important fractionp of the ionizing flux is ab- Ft Fd = Ft + . (17) sorbed in the photoevaporated wind and only a small fraction ξt of the stellar flux arrives at the knot surface. The recombina- In order to obtain the total number of Hα photons emitted by tions in the column beyond Rh mainly balance the stellar flux. If cos( ) F we have the ′′Flux Dominated Regime ′′, the tails of the cometary knots, we assume that the Hα emitting ξh/ θ >> ⋆ ≡ in which : region has a cylindrical section with a thickness h ω Rt . The number of Hα photonsemitted per unit time and per unit length of this cylinder is : Fh(θ) ≈ F⋆ cos(θ), (11)

In this regime almost all the stellar flux arrives at the knot sur- SHα 2 =2 π R ω R n α , (18) face and there is no absorptionin the photoevaporatedflow. The ∆l t t t Hα paticle flux at Rh is roughly equal to the stellar flux. and as a function of the parameter ξt the Hα emission is : In order to compute the total rate of Hα photons emitted by the head of a cometary knot, assuming ω ≪ 1 we have to eval- 2 SHα αHα Ft uate the integral : =2 π Rt , (19) ∆l αB ξt where ∆l is a unit of length along the cylindrical tail. If we π/2 2 2 combine equations (17) and (19) we can calculate the diffuse SHα ≃ 2 π Rh αHα nh(θ) sin(θ) ω Rh dθ , (12) 0 ionizing flux as a function of the Hα emission per unit length Z and the radius of the cylinder : where αHα is the effective Hα recombination coefficient. Taking into account the angular dependence of the density 1/2 SHα/∆l αB SHα/∆l αB profile we can integrate (12) to obtain the Hα emission of the Fd = + ξt . (20) 2 π R α 2 π R α cometary heads : t Hα  t Hα  2.3. Diffuse ionizing field inside an H II region 2 3/2 α α ξ ξ 4F As well as the radial ionizing radiation field from the Helix S ≃ πR2 H F 1 + h − h 1 + ⋆ − 1 . Hα h α ⋆ F 6F2 ξ central star, there will also be a diffuse ionizing radiation field, B ( ⋆ ⋆ " h  #) principally due to ground level recombinations of hydrogen in (13) the nebula, plus smaller contributions from helium recombina- The term in curly brackets is the fraction of incident ionizing tions and the scattering of stellar radiation by dust grains. For photons that are absorbed in the photoevaporating flow before the purposes of calculating the properties of the knot tails, the reaching the ionization front. It is only this fraction of the in- important quantity is the lateral flux of the diffuse field, which cident photons that are reprocessed into Hα radiation. If we is the flux incidenton the surface of an opaque, radially aligned, know the size of the knot Rh and the Hα emission SHα we can thin cylinder. use equation (13) to estimate the stellar flux. In the “on-the spot” (OTS) approximation, in which all dif- 2.2. A model for the cometary tail fuse ionizing photons are assumed to be reabsorbed by neutral H very close to their point of emission, the diffuse flux, Fd, For the cometary tail, we consider the problem of a cylinder across any opaque surface is related to the stellar flux, F∗, at the of neutral material behind the cometary head being photoion- same position by (Henney 2000) ized by the diffuse flux of the nebula. A schematic diagram of this configuration is shown in Figure 2. If we assume that the Fd α1 βOTS ≡ = , (21) radius of the cometary head is much smaller than the distance F∗ 4αBκ to the source we can consider the shadow region to have po- where α and α are, respectively, the H recombination rates lar symmetry. This cylindrical shadow does not receive direct 1 B to the ground level and to all excited levels and κ = σ¯ /σ¯∗, stellar radiation. We therefore have : d where σ¯d is the mean photoionization cross-section averaged over the diffuse ionizing spectrum and σ¯∗ is the same quantity F = F + R n2 (14) d t ω t t αB , averaged over the stellar spectrum. Henney (2000) presents de- where Fd is the diffuse ionizing flux of the surroundings, Ft is tailed calculations, in which the OTS assumption is relaxed, for the incident flux at the tail surface and the second term on the the lateral diffuse flux in two simplified geometries: a classical right represents the absorptions in the photoevaporated wind of filled-sphere homogeneous Strömgren H II region, and a hol- the cometary tail. low cavity H II region with the ionized gas concentrated in a The incident flux at the neutral surface of the tail is related to thin spherical shell. It is found that in both cases the value of (= Fd ) is much smaller than at small radii. It is because the density nt at the base of the cylindrical wind through : β F∗ βOTS 4 PhotoevaporatingflowsfromthecometaryknotsintheHelix nebula (NGC 7293)

Fd

n t F*

R t

Ft Fd

FIG. 2.— Schematic diagram of the cometary tail. Ft is the incident flux at the surface of the neutral cylinder, Fd is the diffuse flux (produced by the surrounding nebula), and Rt and nt are the radius and the density of the neutral tail (respectively).

−2 at small radi, F⋆ gets huge (r ) but Fd, which is proportional to 3. PROPERTIES OF THE COMETARY KNOTS the recombination rate times a path length, plateaus to a more We have used the HST images from the data archive at the or less constant value. In the thin-shell case, β remains at a low Space Telescope Science Institute to obtain the Hα emission value throughout the interior of the cavity, only becoming com- from the cometary knots in the . These images parable to βOTS at the position of the shell. In the filled-sphere are flux-calibrated using the coefficients given in O’Dell & Doi case, on the other hand, for the relatively high values of ap- κ 1999. The f656n and f658n filters was selected to isolated the propriate for the Helix knots (see below), β rises to ≃ β at a OTS Hα and [NII] emission at 658.4 nm. The [NII] emission is not fractional radius of ≃ 0 5 and is roughly constant thereafter. . much stronger than Hα, and thus it does not produce an impor- The stellar ionizing radiation field is much harder than the tant contamination of the Hα observations (less than ≈ 15%). diffuse field, while the photoionization cross-section declines Several methods have been used in order to obtain the dis- rapidly with frequency above the ionization threshold. Hence, tance to this PN (Cahn & Kaler 1971; Daub 1982; Cahn, Kaler the ratio of mean cross-sections, , for the diffuse and stellar κ & Stanghellini 1992; Harris et al 1997), leading to distances fields is larger than unity. Including the H and He0 recombi- ¯ ≃ ranging from 120 to 400 pc. We use the value of 213 pc deter- nation spectrum gives σd 0.75σ0, where σ0 is the threshold mined by Harris et al. (1997) from trigonometric parallax. cross-section and He H=0 13 (O’Dell 1998; Henry, Kwitter / . The comparison between the models and the observations is & Dufour 1999) has been assumed. done for 26 knots, chosen for being relatively well isolated, and The temperatureof the central star of the Helix has been mea- covering a range of distances to the central star. sured by the Hβ Zanstra method to be ≃ 1.2 × 105 K (Górny, Stasinska´ & Tylenda 1997). A further constraint on the stellar spectrum is the observation (O’Dell 1998) that the He++ zone 3.1. Sizes of the knots in the nebula is roughly half the radius of the He+ zone, imply- In order to determine the sizes of the knots, we carry out ing that roughly 10% of the ionizing photons have frequencies aperture photometry with a series of circular diaphragms (cen- higher than the He+ ionization limit. Model atmospheres for tred on the knots) of increasing angular radius. If one plots the compact hot stars (Rauch 1997) that are consistent with these flux within the aperture versus radius, for large enough radii the constraints have photon spectra Lν /hν that are flat or rising flux has to increase as the square of the radius (due to the pres- between the H and He+ ionization limits, implying a value of ence of the bright, surrounding nebular environment). From a σ¯∗ ≃ 0.17σ0. logarithmic plot it is then straightforward to determine the ra- Hence, κ ≃ 4.5 is appropriate for the Helix knots, which dius Rext = Rh + ωRh (see equations 4-5 and Figure 1) at which implies βOTS ≃ 0.033 for an assumed electron temperature the quadratic flux vs. radius dependence first appears. In this 4 of 10 K. The true situation in the Helix nebula is proba- way one then determines the value of the radius Rh of the neu- bly intermediate between the filled-sphere and the hollow-shell tral clump. case. Although O’Dell (1998) finds that the electron density is Applying this method to the 26 chosen knots we find that roughly constant with radius throughout the nebula, the central there is no correlation between the size of the knots and their high-ionization “hole” has a higher temperature and hence a distance to the central star (see Figure 4). However, this could lower emissivity of diffuse ionizing photons. Furthermore, the be a result of the fact that we only have a few knots, and that geometry is more disk-like than spherical, which would tend to these knots are not necessarily representative of the true dis- reduce the intensity of the diffuse field. tribution of knot sizes (as they were chosen so as to be well The radial dependenceof β in the two limiting cases is shown isolated, see above). We calculate a mean radius for the knots ′′ in Figure 3. This is compared with the observational data in of < Rh >= (0.68 ± 0.07) . section 3.3. López-Martín et al. 5

FIG. 3.— Comparison of the Henney (2000) models to the observed values of the diffuse to direct flux ratio (crosses) as a function of the distance from the ionization front. The upper horizontal line is the OTS value and the solid line and dashed line give the variation of β across the nebula for the filled-sphere and hollow-shell case, respectively.

3.2. Hα intensities of the cometary heads with our fit is excellent. Subtracting the background from the fluxes determined with 3.3. Hα intensities of the tails the aperture photometry,we obtain the Hα fluxes emitted by the knots. One can then use equation (13) to calculate the stellar In order to obtain the Hα emission from the tails we integrate ionizing flux at the position of the successive knots as a func- the observed emission over a rectangular area covering the re- tion of distance from the central star. gion in which the tails are clearly detected. We also integrate Column 6 of Table 1 gives the resulting Hα photon pro- the emission in two small adjacent rectangular areas in order duction rates SHα of the knots, using values for ω =0.1 and to determine the intensity of the nebular background, which we −1 10 −2 −1 ci =10km s (therefore ξh ≃ 1.79 × 10 cm s for a knot subtract from the emission of the box containing the cometary ′′ with the mean radius < Rh >=0.68 ). Figure 5 shows SHα vs. tail. Dh (where Dh is the projected distance between the source and Knowing the Hα emission we can then calculate the diffuse the clumps). flux with equation (20) for the 26 chosen knots. In this way we Figure 5 also shows the SHα vs. Dh predicted from equa- calculate the diffuse flux for different distances to the central 2 tion (13) for different values of S∗ (= 4πDh F∗). The predicted star (see column 7 of Table 1). As we have also computed the curves should represent upper envelopes of the observed points, direct stellar flux from the emission of the heads of the knots as the real distances between the clumps and the source are (section 3.2), we can then calculate the diffuse-to-direct stellar larger than the observed, projected distances. From a compar- flux ratios, and compare these values with the ones predicted ison of the predicted curves with the values measured for the from the (Henney 2000) models described in section 2.3. This clumps, we see that the observations can best be fitted with a comparison is shown in Figure 3, where we see that the ob- stellar ionizing photon rate of served points fall into the region between the curves delimited by the two models, indicating that the case of the Helix nebula 45 −1 is intermediate between the homogeneus sphere and the “thin S⋆ ≈ 5 × 10 s . (22) shell” cases. If the knots were distributed isotropically around the central star, then the median knot would have a true distance ≃ 1.15 3.4. Knot masses and evaporation rates times greater than its projected distance. However, the lower From the analysis it is also possible to derive knot masses envelope of the observed knot brightness distribution suggests and evaporation rates. Under the assumption that the knots are ± ◦ that all knots lie within 45 of the plane of the sky, consistent accelerating and have an exponential density profile, one can with a ring-like spatial distribution. derive their mass by According to Osterbrock (1989) S⋆ can be derived from the Hβ luminosity : − 2 16Fc(0)mci 3 M = 1 2 Rc , (24) α L(Hβ) π πcn S = B (23)   ⋆ eff . see Mellema et al. (1998), equation (32). In this equation m αHβ hνHβ is the average mass per atom or ion, for wich we take 1.3mH, If we use the value of the Hβ luminosity of the Helix Nebula Rc is the radius of the clump, and cn is the isothermal velocity obtained by O’Dell (1998) we have a value for the stellar ion- of sound in the neutral gas for which we take 1 km s−1 (corre- izing photon rate of 5.25 × 1045 s−1. As we can see the match sponding to 150 K). Applying this to the measured knots gives 6 PhotoevaporatingflowsfromthecometaryknotsintheHelix nebula (NGC 7293)

TABLE 1 LOCATION AND FLUXES OF THE COMETARY KNOTS 4 a ′′ 5 −1 6 −1 −1 −2 −1 −2 −1 α2000 δ2000 Rh ( ) SHα (s ) SHα (cm s ) F∗ (cm s ) Fd (cm s ) Fd/F⋆ 429 -860 0.73 10.84 × 1039 2.96 × 1022 1.02 × 1010 1.67 × 108 1.63 × 10−2 433 -853 0.69 5.16 × 1039 1.60 × 1022 7.21 × 109 1.29 × 108 1.79 × 10−2 431 -844 0.66 4.67 × 1039 2.53 × 1022 7.53 × 109 1.70 × 108 2.26 × 10−2 440 -848 0.73 4.30 × 1039 – 5.93 × 109 –– 452 -901 0.69 3.44 × 1039 1.54 × 1022 5.74 × 109 1.27 × 108 2.20 × 10−2 428 -827 0.65 2.52 × 1039 2.25 × 1022 5.31 × 109 1.63 × 108 3.07 × 10−2 459 -905 0.67 3.36 × 1039 1.61 × 1022 6.22 × 109 1.33 × 108 2.15 × 10−2 413 -818 0.66 3.39 × 1039 – 6.27 × 109 –– 425 -822 0.71 3.67 × 1039 – 5.93 × 109 –– 352 -815 0.72 7.05 × 1039 2.51 × 1022 8.68 × 109 1.56 × 108 1.79 × 10−2 410 -808 0.68 3.79 × 1039 – 6.67 × 109 –– 474 -931 0.69 6.11 × 1039 1.52 × 1022 7.99 × 109 1.26 × 108 1.58 × 10−2 473 -919 0.64 2.55 × 1039 1.53 × 1022 5.89 × 109 1.36 × 108 2.31 × 10−2 378 -800 0.67 4.26 × 1039 1.58 × 1022 7.13 × 109 1.32 × 108 1.86 × 10−2 354 -804 0.71 5.14 × 1039 1.81 × 1022 7.21 × 109 1.33 × 108 1.85 × 10−2 465 -853 0.67 2.35 × 1039 – 5.10 × 109 –– 480 -925 0.72 2.45 × 1039 – 4.73 × 109 –– 351 -802 0.71 5.32 × 1039 3.16 × 1022 7.35 × 109 1.78 × 108 2.41 × 10−2 398 -752 0.66 3.22 × 1039 1.68 × 1022 6.06 × 109 1.38 × 108 2.28 × 10−2 386 -750 0.65 3.31 × 1039 1.78 × 1022 6.17 × 109 1.68 × 108 2.73 × 10−2 360 -751 0.68 2.79 × 1039 2.62 × 1022 5.61 × 109 1.68 × 108 3.00 × 10−2 352 -750 0.65 3.94 × 1039 – 6.82 × 109 –– 389 -742 0.64 2.74 × 1039 – 6.11 × 109 –– 363 -740 0.64 3.19 × 1039 2.45 × 1022 6.67 × 109 1.73 × 108 2.59 × 10−2 494 -911 0.65 2.77 × 1039 – 5.55 × 109 –– 372 -725 0.65 3.19 × 1039 – 6.06 × 109 ––

FIG. 4.— The angular radii of the knots as a function of projected angular distance to the central star. The crosses indicate the angular sizes determined for individual knots. The horizontal, continuous line indicates the mean value of knot radius, and the dashed horizontal lines give the value of the dispersion from the mean value. The oblique dotted line gives the best fit to the observed data. López-Martín et al. 7

FIG. 5.— Model fits to the Hα photon rates observed for the knots (as a function of distance from the central star) obtained for different stellar ionizing photon rates. The predicted curves should represent upper envelopes of the observed points.

−6 2 an average mass of 9×10 M⊙ with knot masses ranging from Regime” is tevap ≈ Rc ci/cn, so that once a clump start to shrink −6 −5 6×10 M⊙ to 1.8×10 M⊙. In the process one has to calcu- by evaporation, the timescale gets shorter and shorter. The re- late the fraction of the ionizing flux which actually reaches the sults from Mellema et al. (1998) show that the evaporation time heads of the knots, Fc(0)/F∗, which comes out to be about 70 of a knot is given by to 80%. The mass range found is very close to the one derived by 5 ln(1 + η) t = t∗ + , (26) O’Dell & Handron (1996) and Meaburn et al. (1998) from dust evap 6 6η extinction measurements and by Huggins et al. (1992) from CO   measurements, showing that the sizes and shapes of the knots with are entirely consistent with the photoevaporation model. 48 2 ciRc The model also allows an estimate of the current mass loss t∗ = 1 − , (27) rate from the heads of the knots, which is given by π π c2π   n and dM 2 = −Fc(0)mπR , (25) 2α F∗R dt c η = B c , (28) 3πc2 see Mellema et al. (1998), equation (36). This gives an aver- i −9 −1 age of −2.2 × 10 M⊙ yr (ranging between -4.1 and −1.6 × which applied to the measured knots gives an evaporation time −9 −1 10 M⊙ yr ). of 1.1 to 1.2 × 104 years. The time for the clumps to move from close to the star to their present position is about 6 × 103 4. DISCUSSION years. In the “Flux Dominated Regime” the evaporation time 2 The results of the previous sections show that our photoevap- is tevap ≈ Rccicn, so that once a clump start to shrink by evap- oration model is entirely successful in explaining the current oration, the timescale gets shorter and shorter. There is an un- situation of the cometary knots in the Helix nebula. One could certainty in this number since the sound speed in the neutral say that we are using the cometary knots as ‘probes’ for the di- gas is not well defined. Probably, the knot has a range of tem- rect and diffuse UV radiation in the nebula, under the assump- peratures, depending for instance on how far the molecule dis- tion that both the heads and tails are photoevaporating. The sociating photons penetrate. The evaporation time is inversely results from the probes are consistent with the results found by proportional to the temperature, the values quoted being valid other means. for 150 K. Huggins et al. (1992) cite a temperature of 25 K for The question which remains unanswered in this approach is the CO gas in the knots. the origin and final fate of the knots. Starting with the their One consequence of the photoevaporation is that gas is fed final fate, the photoevaporation models give an estimate of the into the region surrounding the knots. Interestingly enough the evaporation time. The evaporation time in the “Flux Dominated Helix Nebula seems to be special among PNe in that its ‘central 8 PhotoevaporatingflowsfromthecometaryknotsintheHelix nebula (NGC 7293) cavity’ is not empty but is at least partly filled with high ioniza- Burkert & O’Dell (1998) proposed that the knots are shaped tion gas producing detectable amounts of [OIII] lines, see for by Lyα photon radiation pressure. This was partly motivated example O’Dell (1998) and Henry et al. (1999). The density by the fact that the evaporation flows appear to have exponen- of this gas is derived to be around 50 cm−3, perhaps less. We tial brightness profiles. These autors found that this mechanism conjecture that this material was injected into the cavity by the does not work in the cometary knots of the Helix nebula. A photoevaporating clouds. Taking a radius of 200′′ for the cav- similar model was proposed for the in the Orion neb- ity, and assuming a spherical shape (probably an overestimate ula (O’Dell 1998). Henney & Arthur (1998) showed that the of the volume and hence of the mass), one finds that the evap- exponential brightness profiles are not inconsistent with photo- −5 oration of 3000 to 5000 knots of about 10 M⊙ can supply evaporation flows and that the radiation pressure from Lyα is this amount of gas. This equals the mass and current number at least one order of magnitude too low to be significant. Their of knots in the Helix. Possibly the gas filling the central cavity arguments also hold for the Helix Nebula cometary knots and is from already evaporated knots, or from the current knots. In we will not repeat them here. The conclusion is that none of the Mellema et al. (1998) it was shown that 50% of the mass of a three models can explain the origin of the long cometary tails photoevaporating clump is lost during the first ‘collapse phase’. under the current circumstances. Since the current knots apparently are in the ‘cometary phase’ When we consider the origin of the knots there are basically they might easily have already lost half of their initial mass. two options, the knots are either primordial or the result of in- The question of the origin of the knots can be divided into stabilities. Meaburn et al. (1998) and Young et al. (1999) favour two questions: “Why do they have a cometary shape?” and the primordialmodel, the main argumentsbeingthat the CO gas “What physical process is responsible for their existence?”. outside of the main nebula has a very clumpy distribution, and Three models exist to explain their shapes. Dyson et al. (1993) that the velocity distribution of the cometary knots follows that showed that collisions between a supersonic wind and a clump of the main CO emission, albeit at a lower expansion velocity. produce short stumpy tails, and that only the interaction be- Within this model it is possible that the tails were shaped as the tween a subsonic wind and a subsonic flow from a clump can knots were run over by the main part of nebula. During this produce long, –like tails. When applying this theorem to phase they would be surrounded and eroded by subsonically our model we run into the problem that the shapes derived in flowing gas, which according to the Dyson et al. (1993) model Dyson et al. (1993) are the shapes of the contact discontinu- would produce long thin tails. Also, the formation of the tails ity between the flow from the clump and the wind, which we as ionization shadows behind the clumps as proposed by Cantó do not trace in our photoionization description. The dynamical et al. (1998) would in this case produce long, dense tails. After models in Mellema et al. (1998) contain both the photoioniza- the main nebular shell has passed by, the knots find themselves tion processes and the interaction of the photoevaporation flow fully exposed to the ionizing stellar radiation and the tails to with the environment. In those models the shape of the contact the diffuse UV radiation field and both start photoevaporating, discontinuity is far from comet–like, as predicted by Dyson et which is the stage in which we see them now. al. (1993); but at the same time the contact discontinuity is not a O’Dell & Handron (1996) prefer the model in which the region which produces a lot of emission, as the ionization front knots are formed as Rayleigh–Taylor instabilities on the inside outshines it by many factors. However, in that model the envi- of the swept up main nebula, proposed originally by Capri- ronment does not exert any ram pressure on the clump and its otti (1973). The numerical model of R–T instabilities presented photoevaporationflow. If the ram pressure in the environment is in figure 4 in O’Dell & Burkert (1997) seems to support this high enough it will overwhelm the photoevaporation flow, and model; it shows elongated structures pointing radially to the dominate the dynamics. Observations show that the environ- centre of the nebula. However, the much higher resolution mod- ment of the cometary knots has a density of about 50 cm−3 and els of Walder & Folini (1998) show the R–T instabilities to be a temperature of perhaps 20,000 K. If the environment is flow- much more chaotic. The fingers in their model need a lot of ing subsonically, this implies pressures of order 10−10 dyn cm−2. processing before they look anything like the cometary knots, The photoevaporationpressure is given by 2Fcmci (see Mellema and one would also expect much more variation in knot prop- et al. 1998),which gives values of about 10−8 dyn cm−2. Clearly erties if they were formed in such instabilities. In all we prefer the environment will not be able to overwhelm the photoevap- the first option of primoridal knots. oration flow from the knots. We cannot rule out the possibility that some time in the 5. CONCLUSIONS past the cometary knots were shaped into their cometary shape In this paper we propose an analytical model for the heads of through a wind–clump interaction. In fact, below we will argue the cometary globules in the Helix Nebula, in which the emis- for such a scenario to explain the origin of the knots. sion is assumed to come from a flow which is being photoe- Cantó et al. (1998) suggested that long tails can be formed vaporated from the surface of a neutral clump. Using the Hα behind clumps opaque to ionizing UV photons. The region be- emission obtained from HST images of the Helix Nebula (ex- hind the clump does not see any direct ionizing photons from tracted from the HST archive, see also O’Dell & Handron1996; the star, just the diffuse UV radiation field, which in general O’Dell & Burkert 1997) and the predictions from our analytic leads to denser, partly neutral tails forming behind the clumps. model, we can calculate the ionizing stellar photon flux at the However, these tails will be initially the same density as the positions of the knots. environment, fill up and recombine, striving towards pressure We find that the knot brightnesses are fully consistent with 45 −1 equilibrium with the ionized environment. The tails of the He- the ionizing photon luminosity of S⋆ =5.25 × 10 s for the lix knots are overpressured compared to their environment, a central star, deduced from the total Hβ flux of the nebula. This situation which does not occur in the scenario put forward in value, however, is somewhat uncertain as in the present work Cantó et al. (1998). The overpressured tails suggest that they we have made no attempt to substract the [N II] 6583 flux from were formed under different circumstances from the ones in the f656n filter. Given the rather strong [N II] emission of some which they are now. of the Helix Knots, our Hα flux determinations could in some López-Martín et al. 9 cases lie up to ≈30 The angular size of the knots has a value of Knowing the direct stellar radiation and the diffuse flux at ′′ < Rh >= (0.68 ± 0.07) (where Rh is the radius of the neutral different distances from the central star (i.e., at the positions clump forming the knot, see section 3.1). From our small sam- of the cometary globules), we can then calculate the diffuse-to- ple of 26 knots, we find no correlation between Rh and distance direct ionizing flux ratio (Fd/F∗) as a function of distance from from the central star (see Figure 4). the central star. We compare the data with simple models for The photoevaporation model also predicts masses for the the diffuse ionizing field (Henney 2000) and conclude that the knots, which come out fully consistent with observationallyde- diffuse field is intermediate in magnitude between that expected rived masses. Their evaporation time is between several thou- from a filled and shell-like geometry, which may be due to the sands to 104 years, dependent on the temperature of the neutral disk-like nature of the Helix nebula. gas in the knots. In this paper we also present a model for the cometary tails as a flow which is being photoevaporated from a neutral, cylin- L. López-Martín is in grateful receipt of a graduate schol- drical “shadow region” behind the neutral clumps which form arship from DGEP-UNAM (México). L. López-Martín, A. C. the knots. This flow is photoevaporated by the impinging dif- Raga and J. Cantó acknowledge support from the CONACyT fuse ionizing field, formed in the surrounding nebula. As we grants 27546-E and 32753-E. W. J. Henney acknowledges sup- have done for the heads of the cometary globules, we compare port from CONACyT grant 27570E and 27546-E, and DGAPA the emission of the cometary tails with the predictions from our grant IN128698. We thank R. C. O’Dell for useful comments model in order to quantitatively determine the value of the dif- and suggestions. We also thank an anonymousreferee for help- fuse ionizing photon flux at the positions of the globules. ful comments.

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